Dr John Brinkley, bishop of Cloyne, is said to have remarked in 1823 of Hamilton at the age of eighteen: “This young man, I do not say will be, but is, the first mathematician of his age.”

William Rowan Hamilton's mathematical included the study of geometrical optics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley-Hamilton Theorem). Hamilton also invented "Icosian Calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.

Hamilton was born in Dublin at 36 Dominick Street. Hamilton showed himself to be a child prodigy. Hamilton was the son of Archibald Hamilton, a solicitor. A branch of the Scottish family to which they belonged had settled in the north of Ireland in the time of James I, and this fact seems to have given rise to the common impression that Hamilton was scottish. Hamilton was educated by James Hamilton (curate of Trim), his uncle and a Anglican priest.

Hamilton's genius first displayed itself in the form of a power of acquiring languages. At the age of seven he had already made very considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle, who was an linguist, almost as many languages as he had years of age. Among these, besides the classical and the modern European languages, were included Persian, Arabic, Hindustani, Sanskrit, and even Malay. But though to the very end of his life he retained much of the singular learning of his childhood and youth, often reading Persian and Arabic in the intervals of sterner pursuits, he had long abandoned them as a study, and employed them merely as a relaxation.

Hamilton was part of a small brilliant school of mathematicians associated with Trinity College, Dublin, where he spent his life. He studied both classics and science, and was appointed Professor of Astronomy in 1827, even before he graduated.

Hamilton's mathematicalal studies seem to have been undertaken and carried to their full development without any assistance whatever, and the result is that his writings belong to no particular “ school,” unless indeed we consider them to form, as they are well entitled to do, a school by themselves. As an arithmetical calculator Hamilton was not only an expert, but he seems to have occasionally found a positive experience in working out to an enormous number of places of decimals the result of some irksome calculation. At the age of twelve Hamilton engaged Zerah Colburn, the American “calculating boy,” who was then being exhibited as a curiosity in Dublin, and he had not always the worst of the encounter. But, two years before, he had accidentally fallen in with a Latin copy of Euclid, which he eagerly devoured; and at twelve Hamilton attacked Newton’s Arithmetica universalis. This was his introduction to modernanalysis. Hamilton soon commenced to read the Principia, and at sixteen Hamilton had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.

About this period Hamilton was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a portion of time to classics. In the summer of 1822, in his seventeenth year, he began a systematic study of Laplace’s Mécanique Céleste. Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; for Laplace’s great work, rich to profusion in analytical processes alike novel and powerful, demands from the student careful and often laborious study.

It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper, “Dês-lors il commença a marcher seul,” to use the words of the biographer of another great mathematician. From that time Hamilton appears to have devoted himself almost wholly to the mathematics investigation, though he ever kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton detected an important defect in one of Laplace’s demonstrations, he was induced by a friend to write out his remarks, that they might be shown to Dr John Brinkley, afterwards bishop of Cloyne, but who was then the first royalastronomer for Ireland, and a accomplished mathematician. Brinkley seems at once to have perceived the vast talents of young Hamilton, and to have encouraged him in the kindest manner.

Hamilton’s career at College was perhaps unexampled. Amongst a number of competitors of more than ordinary merit, he was first in every subject and at every examination. He achieved the rare distinction of obtaining an optime for both Greek and for physics. The amount of many more such honours Hamilton might have attained it is impossible to say; but Hamilton was expected to win both the gold medals at the degree examination, had his career as a student not been cut short by an unprecedented event. This was Hamilton’s appointment to the Andrews professorship of astronomy in the university of Dublin, vacated by Dr Brinkley in 1827. The chair was not exactly offered to him, as has been sometimes asserted, but the electors, having met and talked over the subject, authorized one of their number, who was Hamilton’s personal friend, to urge Hamilton to become a candidate, a step which Hamilton’s modesty had prevented him from taking. Thus, when barely twenty-two, Hamilton was established at the Observatory, Dunsink, near Dublin.

Hamilton was not specially fitted for the post, for although he had a profound acquaintance with theoretical astronomy, he had paid but little attention to the regular work of the practical astronomer. And it must be said that Hamilton’s time was better employed in original investigations than it would have been had he spent it in observations made even with the best of instruments. Hamilton was intended by the university authorities who elected him to the professorship of astronomy to spend his time as Hamilton best could for the advancement of science, without being tied down to any particular branch. If Hamilton devoted himself to practical astronomy, the University of Dublin would assuredly have furnished him with instruments and an adequate staff of assistants.

In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, he was knighted by the lord-lieutenant. But far higher honours rapidly succeeded, among which his election in 1837 to the president’s chair in the Royal Irish Academy, and the rare distinction of being made corresponding member of the academy of St Petersburg. These are the few salient points (other, of course, than the epochs of Hamilton’s more important discoveries and inventions presently to be considered) in the uneventful life of Hamilton.

He made important contributions to optics and to dynamics. Hamilton's papers on optics and dynamics demonstrated theoretical dynamics being treated as a branch of pure mathematics. Hamilton's first discovery was contained in one of those early papers which in 1823 Hamilton communicated to Dr Brinkley, by whom, under the title of “Caustics,” it was presented in 1824 to the Royal Irish Academy. It was referred as usual to a committee Their report, while acknowledging the novelty and value of its contents recommended that, before being published, it should be still further developed and simplified. During the time between 1825 to 1828 the paper grew to an immense bulk, principally by the additional details which had been inserted at the desire of the committee. But it also assumed a much more intelligible form, and the features of the new method were now easily to be seen.

Hamilton himself seems not till this period to have fully understood either the nature or importance of optics, as later Hamilton had intentions of applying his method to dynamics. The Royal Irish Academy paper was finally entitled “Theory of Systems of Rays,” and the first part was printed in 1828 in the Transactions of the Royal Irish Academy. It is understood that the more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers “On a General Method in Dynamics,” which appeared in the Philosophical Transactions in 1834 and 1835.

The principle of “Varying Action“ is the great feature of these papers; and it is, indeed, that the one particular result of this theory which, perhaps more than anything else that Hamilton has done, something which should have been easily within the reach of Augustin Fresnel and others for many years before, and in no way required Hamilton’s new conceptions or methods, although it was by Hamilton’s new theoretical dynamics that he was led to its discovery. This singular result is still known by the name “conical refraction,” which he proposed for it when he first predicted its existence in the third supplement to his “Systems of Rays,” read in 1832.

The step from optics to dynamics in the application of the method of “Varying Action” was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject. These display, like the “Systems of Rays,” a mastery over symbols and a flow of mathematical language almost unequalled. But they contain what is far more valuable still, the greatest addition which dynamical science had received since the strides made by Sir Isaac Newton and Joseph Louis Lagrange. C. G. J. Jacobi and other mathematicians have extended Hamilton’s processes, and have thus made extensive additions to our knowledge of differential equations.

And though differential equations, optics and theoretical dynamics of course are favored in which any such contribution to science can be looked at, the other must not be despised. It is characteristic of most of Hamilton’s, as of nearly all great discoveries, that even their indirect consequences are of high value.

Hamilton discovered quaternions in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. Hamilton could not do so for 3 dimensions, but 4 dimensions produce quaternions. According to the story Hamilton told, on October 16 Hamilton was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

i2 = j2 = k2 = ijk = -1

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians (including Murray Gell-Mann in 2002 and Andrew Wiles in 2003) take a walk from Dunsink observatory to the bridge where, unfortunately no trace of the carving remains.

The quaternion involved abandoning the commutative law, a radical step for the time. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part.

In 1852, Hamilton introduced quaternions as a method of analysis. His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals. Hamilton confidently declared that quaternions would be found have a powerful influence as an instrument of research. Peter Guthrie Tait among others, advocated the use of Hamilton's Quaternions. Quaternions is applicable to concise and elegant demonstrations, it is but seldom used by mathematicians today.

There was controversy about the use of quaternions. Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (from developers like Oliver Heaviside and Willard Gibbs [and vector calculus was later applied to four-vectors]), because quaternions provide superior notation. While this is undebatable in four dimensions, quaternions cannot be used with arbitrary dimensionality (though extensions like Octonions and Clifford algebras may be more applicable). Vector notation has replaced the "space-time" quaternions in science and engineering by the mid-20th century.

Hamilton proceeded to popularize quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death. Today, the quaternions are in use by computer graphics, control theory, signal processing and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.

Hamilton also contributed an alternative formulation of the mathematical theory of classical mechanics. While adding no new physics, this formulation, which builds on that of Joseph Louis Lagrange, provides a more powerful technique for working with the equations of motion. Both the Lagrangian and Hamiltonian approaches were developed to describe the motion of discrete systems, were then extended to continuous system and in this form can be used to define fieldss. In this way, the techniques find use in electromagnetic, quantum and relativity theory.

Hamilton's extraordinary investigations connected with the solution of algebraic equations of the fifthdegree, and his examination of the results arrived at by N. H. Abel, G. B. Jerrard, and others in their researches on this subject, form another contribution to science. There is next Hamilton's paper on Fluctuating Functions, a subject which, since the time of J. Fourier, has been of immense and ever increasing value in physical applications of mathematics. There is also the extremely ingenious invention of the hodograph. Of his extensive investigations into the solutions (especially by numerical approximation) of certain classes of physical differential equations, only a few items have been published, at intervals, in the PhilosophicalMagazine.

Besides all this, Hamilton was a voluminous correspondent. Often a single letter of Hamilton's occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of Hamilton's mind never to be satisfied with a general understanding of a question; Hamilton pursued the problem until he knew it in all its details. Hamilton was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much time. He was excessively precise and hard to please with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little compared with the extent of Hamilton's investigations.

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